■同じ円に内接するいくつかの正多角形(その4)

 (その3)の続き.

  cosπ/7+cos3π/7+cos5π/7=1/2

=(a^2+b^2+c^2)/2R^2-3=1/2

  a^2+b^2+c^2=7R^2

  cosπ/7・cos3π/7+cos3π/7・cos5π/7+cos5π/7・cosπ/7=-1/2

  (c^2/2R^2-1)(b^2/2R^2-1)+(b^2/2R^2-1)(a^2/2R^2-1)+(a^2/2R^2-1)(c^2/2R^2-1)=-1/2

  (c^2-2R^2)(b^2-2R^2)+(b^2-2R^2)(a^2-2R^2)+(a^2-2R^2)(c^2-2R^2)=-2R^4

  a^2b^2+b^2c^2+c^2a^2-4R^2(a^2+b^2+c^2)+12R^4=-2R^4

  a^2b^2+b^2c^2+c^2a^2=14R^4

  cosπ/7・cos3π/7・cos5π/7=-1/8

  (c^2/2R^2-1)(b^2/2R^2-1)(a^2/2R^2-1)=-1/8

  (c^2-2R^2)(b^2-2R^2)(a^2-2R^2)=-R^6

  a^2b^2c^2-2R^2(b^2c^2+c^2a^2+a^2b^2)+4R^4(a^2+b^2+c^2)-8R^6=-R^6

  a^2b^2c^2-28R^6+28R^6-8R^6=-R^6

  a^2b^2c^2=7R^6

  abc=√7R^3

 1/a^2+1/b^2+1/c^2

=(a^2b^2+b^2c^2+c^2a^2)/a^2b^2c^2=14/7R^2=2/R^2

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